My reply is reference

Regions of a Circle Cut by Chords to n Points
———————————————- n factors are distributed around the circumference of a circle and every level is
joined to each different level by a chord of the circle. Assuming that
no three chords intersect at some extent contained in the circle we require the
variety of areas into which the circle is split.

With no traces the circle has only one area. Now take into account any
assortment of traces. If you draw a brand new line throughout the circle which
doesn’t cross any current traces, then the impact is to extend the
variety of areas by 1. In addition, each time a brand new line crosses an
current line contained in the circle the variety of areas is elevated by
1 once more.

So in any such association
variety of areas = 1 + variety of traces + variety of inside
intersections

= 1 + C(n,2) + C(n,4)
Note that the variety of traces is the variety of methods 2 factors could be
chosen from n factors. Also, the variety of inside intersections is
the variety of quadrilaterals that may be shaped from n factors, since
every quadrilateral produces simply 1 intersection the place the diagonals
of the quadrilateral intersect.
Examples:
n=4 Number of areas = 1 + C(4,2) + C(4,4) = 8
n=5 Number of areas = 1 + C(5,2) + C(5,4) = 16
n=6 ” ” = 1 + C(6,2) + C(6,4) = 31
n=7 ” ” = 1 + C(7,2) + C(7,4) = 57